Editing Involution (mathematics) (section)

=== Group theory ===
In [[group theory]], an element of a [[group (mathematics)|group]] is an involution if it has [[order (group theory)|order]] 2; that is, an involution is an element {{math|''a''}} such that {{math|''a'' ≠ ''e''}} and {{math|1=''a''<sup>2</sup> = ''e''}}, where {{math|''e''}} is the [[identity element]].<ref>
John S. Rose.
[https://books.google.com/books?id=j-I7Zpq3GdIC "A Course on Group Theory"].
p. 10, section 1.13.
</ref>
Originally, this definition agreed with the first definition above, since members of groups were always bijections from a set into itself; that is, ''group'' was taken to mean ''[[permutation group]]''. By the end of the 19th century, ''group'' was defined more broadly, and accordingly so was ''involution''.

A [[permutation]] is an involution if and only if it can be written as a finite product of disjoint [[transposition (mathematics)|transposition]]s.

The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the [[classification of finite simple groups]].

An element {{math|''x''}} of a group {{math|''G''}} is called [[strongly real element|strongly real]] if there is an involution {{math|''t''}} with&nbsp;{{math|1=''x''{{sup|''t''}} = ''x''{{sup|−1}}}} (where&nbsp;{{math|1=''x''{{sup|''t''}} = ''x''{{sup|−1}} = ''t''{{sup|−1}} ⋅ ''x'' ⋅ ''t''}}).

[[Coxeter group]]s are groups generated by a set {{math|''S''}} of involutions subject only to relations involving powers of pairs of elements of {{math|''S''}}.  Coxeter groups can be used, among other things, to describe the possible [[Platonic solid|regular polyhedra]] and their [[regular polytope|generalizations to higher dimensions]].